Eur. J. Mech. A/Solids 18 (1999) 939–962  1999 Éditions scientifiques et médicales Elsevier SAS. All rights reserved On coupled gradient-dependent plasticity and damage theories with a view to localization analysis René de Borst a , 1 , Jerzy Pamin b , Marc G.D. Geers c a Koiter Institute Delft/Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands b Faculty of Aerospace Engineering, Delft University of Technology/Faculty of Civil Engineering, Cracow University of Technology, Poland c Faculty of Mechanical Engineering, Eindhoven University of Technology/Faculty of Civil Engineering, Royal Military Academy, Brussels, Belgium (Received 12 November 1998; revised and accepted 3 February 1999) Abstract – Combinations of gradient plasticity with scalar damage and of gradient damage with isotropic plasticity are proposed and implemented within a consistently linearized format. Both constitutive models incorporate a Laplacian of a strain measure and an internal length parameter associated with it, which makes them suitable for localization analysis. The theories are used for finite element simulations of localization in a one-dimensional model problem. The physical relevance of coupling hardening/softening plasticity with damage governed by different damage evolution functions is discussed. The sensitivity of the results with respect to the discretization and to some model parameters is analyzed. The model which combines gradient-damage with hardening plasticity is used to predict fracture mechanisms in a Compact Tension test.  1999 Éditions scientifiques et médicales Elsevier SAS gradient-dependent continuum / plasticity / damage / strain localization / finite elements 1. Introduction The problem of strain localization driven by material instabilities has been thoroughly investigated, see for instance (de Borst et al., 1993) or the recent book edited by de Borst and van der Giessen (1998), and is by now rather well understood. If a material instability (Hill, 1958) is encountered in the deformation history of a body, the strains tend to localize in a number of narrow bands, while the remaining parts of the body unload. Within a classical, local continuum formulation this phenomenon is, for static problems, associated with the loss of ellipticity of the governing partial differential equations, and therefore, discretization methods used to solve them may yield meaningless results. To overcome this problem, some form of rate-dependent or nonlocal enhancement of the constitutive model must be adopted (de Borst et al., 1993). In other words, a continuum formulation equipped with an internal length parameter should be used. As constitutive framework, a combination of plasticity and damage theories is physically appealing since a host of materials exhibit an interaction of inelastic mechanisms of microcrack or microvoid growth with plastic flow. The coupled models are more flexible and make it possible to reproduce a realistic elastic stiffness degradation, which is important for cyclic loading and extensive stress redistributions. Within a local continuum format, plasticity and damage couplings have been analyzed in a small strain format by Simo and Ju (1987), Ju (1989), Hansen and Schreyer (1994), Doghri (1995), and in a large strain format by 1 E-mail: r.deborst@lr.tudelft.nl.